
As before, with sufficient probability we have \(d^{2}/|I|^2 \lesssim n^{-2}d^3\) . The only difficulty is that we need a slightly sharper variant of the upper bound on \(\Vert A - \operatorname{\mathbb {E}}A\Vert \) . Recalling the bound (REF ), by the matrix Bernstein inequality [1]} we have
\(\Pr \left({A - \operatorname{\mathbb {E}}A} \ge {I}d^{-3/2}\right)\le d \exp \left(-\frac{{I}^{2}d^{-3}/2}{4{I}d^{-1/2} + d/3}\right),\) 
